Skip to main content
SpringerLink
Log in

Classes of Infinitely Divisible Distributions and Examples

  • Chapter
Lévy Matters V

Part of the book series: Lecture Notes in Mathematics ((LEVY,volume 2149))

  • 1109 Accesses

Abstract

Bondesson (Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statistics, vol. 76, Springer, Berlin, 1992) said “Since a lot of the standard distributions now are known to be infinitely divisible, the class of infinitely divisible distributions has perhaps partly lost its interest. Smaller classes should be more in focus.” This view was presented more than two decades ago, yet has not been fully addressed. Over the last decade, many classes of infinitely divisible distributions have been studied and characterized. In this article, we summarize such “smaller classes” and try to find classes which known infinitely divisible distributions belong to, as precisely as possible.

AMS Subject Classification 2000: Primary: 60E07, 60H05; Secondary: 60E10, 60G51

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. K. Akita, M. Maejima, On certain self-decomposable self-similar processes with independent increments. Stat. Probab. Lett. 59, 53–59 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Alf, T.A. O’Connor, Unimodality of the Lévy spectral function. Pac. J. Math. 69, 285–290 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Aoyama, M. Maejima, Characterizations of subclasses of type G distributions on \(\mathbb{R}^{d}\) by stochastic integral representations. Bernoulli 13, 148–160 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Aoyama, M. Maejima, J. Rosiński, A subclass of type G selfdecomposable distributions on \(\mathbb{R}^{d}\). J. Theor. Probab. 21, 14–34 (2008)

    Article  MATH  Google Scholar 

  5. T. Aoyama, A. Lindner, M. Maejima, A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class. Electron. J. Probab. 15, 1119–1142 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Aoyama, M. Maejima, Y. Ueda, Several forms of stochastic integral representations of gamma random variables and related topics. Probab. Math. Stat. 32, 99–118 (2011)

    MathSciNet  Google Scholar 

  7. O.E. Barndorff-Nielsen, S. Thorbjørnsen, Lévy laws in free probability. Proc. Natl. Acad. Sci. U.S.A. 99, 16568–16575 (2002)

    Article  MATH  Google Scholar 

  8. O.E. Barndorff-Nielsen, S. Thorbjørnsen, Lévy processes in free probability. Proc. Natl. Acad. Sci. U.S.A. 99, 16576–16580 (2002)

    Article  MATH  Google Scholar 

  9. O.E. Barndorff-Nielsen, S. Thorbjørnsen, A connection between free and classical infinitely divisibility. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 573–590 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. O.E. Barndorff-Nielsen, S. Thorbjørnsen, Regularising mapping of Lévy measures. Stoch. Process. Appl. 16, 423–446 (2006)

    Article  Google Scholar 

  11. O.E. Barndorff-Nielsen, J. Kent, M. Sørensen, Normal variance-mean mixtures and z distributions. Int. Stat. Rev. 50, 145–159 (1982)

    Article  MATH  Google Scholar 

  12. O.E. Barndorff-Nielsen, M. Maejima, K. Sato, Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–33 (2006)

    MathSciNet  MATH  Google Scholar 

  13. J. Bertoin, M. Yor, On the entire moments of self-similar Markov processes and exponential functionals. Ann. Fac. Sci. Toulouse Math. VI 11, 33–45 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Bertoin, T. Fujita, B. Roynette, M. Yor, On a particular class of self decomposable random variables: the duration of a Bessel excursion straddling an independent exponential time. Probab. Math. Stat. 26, 315–366 (2006)

    MathSciNet  MATH  Google Scholar 

  15. J. Bertoin, A. Lindner, R. Maller, On continuity properties of the law of integrals of Lévy processes, in Séminaire de Probabilités XLI, ed. by C. Donati-Martin et al. Lecture Notes in Mathematics, vol. 1934 (Springer, Berlin, 2008), pp. 137–159

    Google Scholar 

  16. L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics, vol. 76 (Springer, Berlin, 1992)

    Google Scholar 

  17. P. Bosch, T. Simon, On the self-decomposability of the Fréchet distribution. Indag. Math. 24, 626–636 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Carr, H. Geman, D.B. Madan, M. Yor, The fine structure of asset returns: an empirical investigation. J. Bus. 75, 305–333 (2002)

    Article  Google Scholar 

  19. P. Carr, H. Geman, D.B. Madan, M. Yor, Pricing options on realized variance. Finance Stochast. 9, 453–475 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. R.L. Dobrushin, P. Major, Non-central limit theorem for non-linear functions of Gaussian fields. Zeit. Wahrschein. verw. Geb. 50, 27–52 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Feller, An Introduction to Probability Theory and Its Applications, vol. II (Wiley, New York, 1971)

    MATH  Google Scholar 

  22. K. Fukuyama, S. Takahashi, On limit distributions of trigonometric sums. Rev. Roum. Math. Pure Appl. 53, 19–24 (2008)

    MathSciNet  MATH  Google Scholar 

  23. C. Goldie, A class of infinitely divisible random variables. Proc. Camb. Philos. Soc. 63, 1141–1143 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Handa, The sector constants of continuous state branching processes with immigration. J. Funct. Anal. 262, 4488–4524 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Ichifuji, M. Maejima, Y. Ueda, Fixed points of mappings of infinitely divisible distributions on \(\mathbb{R}^{d}\). Stat. Probab. Lett. 80, 1320–1328 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. A.M. Iksanov, Z.J. Jurek, B.M. Schreiber, A new factorization property of the selfdecomposable probability measures. Ann. Probab. 32, 1356–1369 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. M.E.H. Ismail, D.H. Kelker, Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10, 884–901 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. L.F. James, B. Roynette, M. Yor, Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5, 346–415 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Jeanblanc, J. Pitman, M. Yor, Self-similar processes with independent increments associated with Lévy and Bessel processes. Stoch. Process. Appl. 100, 223–231 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Z.J. Jurek, Limit distributions for sums of shrunken random variables. Diss. Math. 185, 1981 (1981)

    MathSciNet  Google Scholar 

  31. Z.J. Jurek, Relations between the s-selfdecomposable and selfdecomposable measures. Ann. Probab. 13, 592–608 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  32. Z.J. Jurek, Random integral representations for classes of limit distributions similar to Lévy class \(L_{0}\). Probab. Theory Relat. Fields 78, 473–490 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Z.J. Jurek, Random integral representations for classes of limit distributions similar to Lévy class \(L_{0}\). II. Nagoya Math. J. 114, 53–64 (1989)

    MathSciNet  MATH  Google Scholar 

  34. Z.J. Jurek, Random integral representations for classes of limit distributions similar to Lévy class \(L_{0}\). III, in Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Bowdoin College in Summer of 1991, ed. by R.M. Dudley et al. (Birkhäuser, Boston, MA, 1992), pp. 137–151

    Google Scholar 

  35. Z.J. Jurek, The random integral representation hypothesis revisited: new classes of s-selfdecomposable laws, in Abstract and Applied Analysis (World Scientific, Singapore, 2004), pp. 479–498

    Google Scholar 

  36. Z.J. Jurek, The random integral representation conjecture: a quarter of a century later. Lith. Math. J. 51, 362–369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Z.J. Jurek, B.M. Schreiber, Fourier transforms of measures from the classes \(\mathcal{U}_{\beta },\ -2 <\beta \leq -1\). J. Multivar. Anal. 41, 194–211 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Z.J. Jurek, W. Vervaat, An integral representation for selfdecomposable Banach space valued random variables. Zeit. Wahrschein. verw. Geb. 62, 247–262 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  39. H. Kondo, M. Maejima, K. Sato, Some properties of exponential integrals of Lévy processes and examples. Electron. Commun. Probab. 11, 291–303 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. L. Lindner, M. Maejima, Unpublished note, Marburg University (2007)

    Google Scholar 

  41. J.V. Linnik, I.V. Ostrovskii, Decomposition of Random Variables and Vectors (American Mathematical Society, Providence, RI, 1977)

    MATH  Google Scholar 

  42. D.B. Madan, E. Seneta, The variance gamma (VG) model for share market returns. J. Bus. 63, 511–524 (1990)

    Article  Google Scholar 

  43. M. Maejima, Subclasses of Goldie-Steutel-Bondesson class of infinitely divisible distributions on \(\mathbb{R}^{d}\) by \(\varUpsilon\)-mapping. Lat. Am. J. Probab. Math. Stat. 3, 55–66 (2007)

    MathSciNet  MATH  Google Scholar 

  44. M. Maejima, G. Nakahara, A note on new classes of infinitely divisible distributions on \(\mathbb{R}^{d}\). Electron. Commun. Probab. 14, 358–371 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Maejima, Y. Niiyama, The generalized Langevin equation and an example of type G distributions. Inst. Stat. Math. Coop. Res. Rep. 175, 126–129 (2005)

    Google Scholar 

  46. M. Maejima, J. Rosiński, The class of type G distributions on \(\mathbb{R}^{d}\) and related subclasses of infinitely divisible distributions. Demonstratio Math. 34, 251–266 (2001)

    MathSciNet  MATH  Google Scholar 

  47. M. Maejima, J. Rosiński, Type G distributions on \(\mathbb{R}^{d}\). J. Theor. Probab. 15, 323–341 (2002)

    Article  MATH  Google Scholar 

  48. M. Maejima, K. Sato, The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions. Probab. Theory Relat. Fields 145, 119–142 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Maejima, C.A. Tudor, On the distribution of the Rosenblatt process. Stat. Probab. Lett. 83, 1490–1495 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Maejima, Y. Ueda, Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes. Commun. Stoch. Anal. 3, 349–367 (2009)

    MathSciNet  Google Scholar 

  51. M. Maejima, Y. Ueda, A note on a bivariate gamma distribution. Stat. Probab. Lett. 80, 1991–1994 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Maejima, Y. Ueda, α-Selfdecomposable distributions and related Ornstein-Uhlenbeck type processes. Stoch. Process. Appl. 120, 2363–2389 (2010)

    Google Scholar 

  53. M. Maejima, Y. Ueda, Compositions of mappings of infinitely divisible distributions with applications to finding the limits of some nested subclasses. Electron. Commun. Probab. 15, 227–239 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Maejima, Y. Ueda, Nested subclasses of the class of α-selfdecomposable distributions. Tokyo J. Math. 34, 383–406 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. M. Maejima, Y. Ueda, Examples of α-selfdecomposable distributions. Stat. Probab. Lett. 83, 286–291 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  56. M. Maejima, K. Sato, T. Watanabe, Distributions of selfsimilar and semi-selfsimilar processes with independent increments. Stat. Probab. Lett. 47, 395–401 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  57. M. Maejima, M. Matsui, M. Suzuki, Classes of infinitely divisible distributions on \(\mathbb{R}^{d}\) related to the class of selfdecomposable distributions. Tokyo J. Math. 33, 453–486 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  58. M. Maejima, V. Pérez-Abreu, K. Sato, A class of multiple infinitely divisible distributions related to arcsine density. Bernoulli 18, 476–795 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. M. Maejima, J. Rosiński, Y. Ueda, Stochastic integral and series representations for strictly stable distributions. J. Theor. Probab. (2015). doi:10.1007/s10959-013-0518-8. Published online: 27 September 2013

    Google Scholar 

  60. I. Nourdin, G. Peccati, Normal Approximations with Malliavin Calculus (Cambridge University Press, Cambridge, 2012)

    Book  MATH  Google Scholar 

  61. T.A. O’Connor, Infinitely divisible distributions similar to class L distributions. Z. Wahrsch. Verw. Gebiete 50, 265–271 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  62. T.A. O’Connor, Infinitely divisible distributions with unimodal Levy spectral functions. Ann. Probab. 7, 494–499 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  63. T.A. O’Connor, Some classes of limit laws containing the stable distributions. Z. Wahrsch. Verw. Gebiete 55, 25–33 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  64. V. Pipiras, M.S. Taqqu, Regularization and integral representations of Hermite processes. Stat. Probab. Lett. 80, 2014–2023 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  65. N. Privault, D. Yang, Infinite divisibility of interpolated gamma powers. J. Math. Anal. Appl. 405, 373–387 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  66. N. Privault, D. Yang, Variance-GGC asset price models and their sensitivity analysis. Preprint (2015), ntu.edu.sg/home/nprivault

    Google Scholar 

  67. A. Rocha-Arteaga, K. Sato, Topics in Infinitely Divisible Distributions and Lévy Processes. Aportaciones Matemáticas, Investigación 17 (Sociedad Matemática Mexicana, Mexico, 2003)

    Google Scholar 

  68. J. Rosinski, On a class of infinitely divisible processes represented as mixtures of Gaussian processes, in Stable Processes and Related Topics, ed. by S. Cambanis et al. (Birkhäuser, Basel, 1991), pp. 27–41

    Chapter  Google Scholar 

  69. J. Rosiński, Tempering stable processes. Stoch. Process. Appl. 117, 677–707 (2007)

    Article  MATH  Google Scholar 

  70. K. Sato, Class L of multivariate distributions and its subclasses. J. Multivar. Anal. 10, 207–232 (1980)

    Article  MATH  Google Scholar 

  71. K. Sato, Self-similar processes with independent increments. Probab. Theory Relat. Fields 89, 285–300 (1991)

    Article  MATH  Google Scholar 

  72. K. Sato, Time evolutions of Lévy processes, in Trend in Probability and Related Analysis, ed. by N. Kôno, N.-R. Shieh (World Scientific, Singapore, 1997), pp. 35–82

    Google Scholar 

  73. K. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999)

    MATH  Google Scholar 

  74. K. Sato, Monotonicity and non-monotonicity of domains of stochastic integral operators. Probab. Math. Stat. 26, 23–39 (2006)

    MATH  Google Scholar 

  75. K. Sato, Two families of improper stochastic integrals with respect to Lévy processes. Lat. Am. J. Probab. Math. Stat. 1, 47–87 (2006)

    MATH  Google Scholar 

  76. K. Sato, Transformations of infinitely divisible distributions via improper stochastic integrals. Lat. Am. J. Probab. Math. Stat. 3, 67–110 (2007)

    MATH  Google Scholar 

  77. K. Sato, Fractional integrals and extensions of selfdecomposability, in Lévy Matters I, ed. by T. Duquesne et al. Lecture Notes in Mathematics, vol. 2010 (Springer, Berlin, 2010), pp. 1–91

    Google Scholar 

  78. K. Sato, Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions. Lat. Am. J. Probab. Math. Stat. 8, 1–17 (2011)

    MATH  Google Scholar 

  79. K. Sato, M. Yamazato, Stationary processes of Ornstein-Uhlenbeck type, in Probability Theory and Mathematical Statistics, ed. by K. Ito, J.V. Prokhorov. Lecture Notes in Mathematics, vol. 1021 (Springer, Berlin, 1983), pp. 541–551

    Google Scholar 

  80. R.L. Schilling, R. Song, Z. Vondra\(\check{c}\) ek, Bernstein Functions. Theory and Applications (De Gruyter, New York, 2010)

    Google Scholar 

  81. P.R. Shanbhag, M. Sreehari, On certain self-decomposable distributions. Z. Wahrsch. verw. Gebiete 38, 217–222 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  82. F.W. Steutel, Note on the infinitely divisibility of exponential mixtures. Ann. Math. Stat. 38, 1303–1305 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  83. F.W. Steutel, K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line (Dekker, New York, 2004)

    MATH  Google Scholar 

  84. T. Takemura, M. Tomisaki, Lévy measure density corresponding to inverse local time. Publ. RIMS Kyoto Univ. 49, 563–599 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  85. O. Thorin, On the infinite divisibility of the Pareto distribution. Scand. Actuar. J. 1977, 31–40 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  86. O. Thorin, On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 1977, 121–148 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  87. O. Thorin, Proof of a conjecture of L. Bondesson concerning infinite divisibility of powers of a gamma variable. Scand. Actuar. J. 1978, 151–164 (1978)

    MathSciNet  MATH  Google Scholar 

  88. C.A. Tudor, Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12, 230–257 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  89. C.A. Tudor, F. Viens, Variations and estimators for the selfsimilarity order through Malliavin calculus. Ann. Probab. 6, 2093–2134 (2009)

    Article  MathSciNet  Google Scholar 

  90. K. Urbanik, Limit laws for sequences of normed sums satisfying some stability conditions, in Multivariate Analysis–III, ed. by P.R. Krishnaiah (Academic, New York, 1973), pp. 225–237

    Google Scholar 

  91. N. Van Thu, Universal multiply self-decomposable probability measures on Banach spaces. Probab. Math. Stat. 3, 71–84 (1982)

    MATH  Google Scholar 

  92. N. Van Thu, Fractional calculus in probability. Probab. Math. Stat. 3, 173–189 (1984)

    Google Scholar 

  93. N. Van Thu, An alternative approach to multiple self-decomposable probability measures on Banach spaces. Probab. Theory Relat. Fields 72, 35–54 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  94. M.S. Veillette, M.S. Taqqu, Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19, 982–1005 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  95. D. Vere-Jones, The infinite divisibility of a bivariate gamma distribution. Sankhyā Ser. A 29, 421–422 (1967)

    MathSciNet  MATH  Google Scholar 

  96. S.J. Wolfe, On a continuous analogue of the stochastic difference equation X n = ρ X n−1 + B n . Stoch. Process. Appl. 12, 301–312 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  97. M. Yamazato, Unimodality of infinitely divisible distribution functions of class L. Ann. Probab. 6, 523–531 (1978)

    Google Scholar 

Download references

Acknowledgements

I would like to express my sincere gratitude to people who collaborated on the papers cited in this article, Ole E. Barndorff-Nielsen, Alexander Lindner, Muneya Matsui, Víctor Pérez-Abreu, Jan Rosiński, Ken-iti Sato, Ciprian A. Tudor, and my students at Keio University. Special thanks to Ken-iti Sato, who first led me to this interesting subject in the theory of infinitely divisible distributions and also read the first draft of this article, giving me many valuable comments. My former Ph.D. student, Yohei Ueda, also deserves special mention for working with me for many years on this topic at Keio University. I would like to acknowledge as well an anonymous referee for suggestions that helped to enhance this article.

Author information

Authors and Affiliations

  1. Department of Mathematics, Keio University, Hiyoshi, Yokohama, 223-8522, Japan

    Makoto Maejima

Authors
  1. Makoto Maejima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Makoto Maejima .

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Maejima, M. (2015). Classes of Infinitely Divisible Distributions and Examples. In: Lévy Matters V. Lecture Notes in Mathematics(), vol 2149. Springer, Cham. https://doi.org/10.1007/978-3-319-23138-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation